From: Thomas Walker Lynch Date: Sat, 18 Jul 2026 11:37:14 +0000 (+0000) Subject: restores the lost theoretic TTCA machine sections X-Git-Url: https://git.reasoningtechnology.com/%5B%5E?a=commitdiff_plain;h=88335db3692e2431e3dab57b371ff21525d511d6;p=TM-2026 restores the lost theoretic TTCA machine sections --- diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index ece380b..1d765c6 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -564,6 +564,143 @@

There is a mechanical procedure for converting a Moore Machine into a Mealy Machine, and the reverse. Two such converted machines are equally expressive. Making the read operation an explicit action, instead of having it implied by a state transition, increases the number of states in a controller, and consequently the number of steps that must be taken. However, the new machine can do in two steps anything the former machine could do in one; consequently, this change affects the multiplier constant on the linear term of the step count formula but does not change the computation complexity class determined from such a step count. The layers of next-state functions can be collapsed into one layer, where the missing arguments are filled in with all possible unused values. This potentially increases the number of state transition arcs that must be specified, but the state transition logic and the number of states remain unchanged. Partitioning the machine definition by separation of concerns does not change the total specification. Hence, these modifications are computationally inconsequential.

+

The TTCA Turing Machine fixed part

+ +
+ + MF = (QF, ΣF, AF, δF_0) + +
+ +

In the following the middle dot acts as a namespace operator, N{·}x. By doing this we assure there will be no aliasing with the symbols provided by the programmer when he defines the programmed state controller.

+ +

The set of predefined states:

+
+ QF = \{QF{·}\mathtt{initial}, QF{·}\mathtt{unspecified\_decision}\} +
+ +

The state controller always starts in the QF·initial state. This is a symbol representing the state, it is not a register that holds a state. The 'F' values are all fixed.

+ +

If a decision is attempted to be made on an ΣF·unspecified symbol, the machine transitions to the QF·unspecified_decision state, due to the global default transition defined in δF_0.

+ +

The set of available actions:

+
+ + \begin{aligned} + AF = \{\\ + & \mathtt{no\_op} \\ + , & \mathtt{left} \\ + , & \mathtt{right} \\ + , & \mathtt{read\_d} \\ + , & \mathtt{read\_g} \\ + , & \mathtt{status} \\ + , & \mathtt{write\_σ}(σ) \\ + , & \mathtt{write\_d} \\ + , & \mathtt{write\_g} \\ + \} + \end{aligned} + +
+

where σ must be in Σ.

+ +

The set of predefined symbols:

+
+ ΣF = \{ΣF{·}\mathtt{unspecified}, ΣF{·}\mathtt{on\_leftmost}\} +
+ +

The set δF_0 holds one global default predefined transition of the form, +

+ + \{\langle \mathtt{g{.}ΣF{·}unspecified}, \mathtt{QF{·}unspecified\_decision} \rangle\} + +
+ +

In any state if the g register holds the symbol \mathtt{ΣF{·}unspecified}, then the next state will be \mathtt{QF{·}unspecified\_decision}, independent of all other defined next state transitions.

+ + +

The TTCA Turing Machine variables

+ +
+ + MV = (q, d, g) + +
+ +

q: is the current state of the machine.

+ +

d: is the data register.

+ +

g: is the gate register.

+ + +

The TTCA Turing Machine programmable part

+ +
+ + MP = (QP, ΣP, λP, δ_1, δ_2, δ_3) + +
+ +

A set of programmed state symbols:

+
+ QP +
+ +

A set of programmed data symbols:

+
+ ΣP +
+ +

The programmed actions. A set of pairs of the form:

+
+ λP = \{ \langle q_0, a \rangle, \dots \} +
+

where q_0 is matched to the current state, and a is a member of AF.

+ +

A set of state transition triples; each triple is of the form:

+
+ δ_1 = \{ \langle q_0, r{·}σ, q_1 \rangle, \dots \} +
+

where q_0 is matched to the current state of the machine. r{·}σ is a symbol in register r, where r is either d or g. q_1 is the next state. When q_0 matches the current state, q_1 becomes the next state. Both q_0 and q_1 come from the total set Q. σ comes from the total set Σ.

+ +

The default transition table. A set of state transition pairs; each pair is of the form:

+
+ δ_2 = \{ \langle q_0, q_1 \rangle, \dots \} +
+

where q_0 is matched to the current state, and upon a match q_1 will be taken as the next state.

+ +

A default next state:

+
+ δ_3 = q_1 +
+ +

This is the transition of last resort. It is unconditional, the next state becomes q_1.

+ +

The TTCA Turing Machine in total

+ +
+ + M = (Q, Σ, AF, λP, δ) + +
+ +

The complete set of states, uniting the fixed predefined states and the programmed states:

+
+ Q = QF \cup QP +
+ +

The complete set of symbols, uniting the fixed control symbols and the programmed data symbols:

+
+ Σ = ΣF \cup ΣP +
+ +

The ordered sequence of next state transition rules:

+
+ δ = [δF_0 | δ_1 | δ_2 | δ_3] +
+ + +

TTCA computation theoretic TM executor directions

@@ -574,14 +711,14 @@

Phase 1: The action

During the action phase, the executor looks up the operation mapped to the current state.

  1. Locate the current state q within the programmed actions set λP.
    2. -
  2. Perform the associated action a \in AF. This will instruct the machine to move the head (left, right), read the symbol under the head into a designated register (read_d, read_g), write a symbol to the tape (write_σ, write_d, write_g), perform a status check, or execute a no_op.
    3. +
  3. Perform the associated action a \in AF.

Phase 2: The state transition